Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.

Let [Formula: see text] be an integer. From a set of [Formula: see text]-dimensional vectors, we obtain a [Formula: see text]-dot product graph by letting each vector [Formula: see text] correspond to a vertex [Formula: see text] and by adding an edge between two vertices [Formula: see text] and [Formula: see text] if and only if their dot product [Formula: see text], for some fixed, positive threshold [Formula: see text]. Dot product graphs can be used to model social networks. Recognizing a [Formula: see text]-dot product graph is known to be [Formula: see text]-hard for all fixed [Formula: see text]. To understand the position of [Formula: see text]-dot product graphs in the landscape of graph classes, we consider the case [Formula: see text], and investigate how [Formula: see text]-dot product graphs relate to a number of other known graph classes including a number of well-known classes of intersection graphs.

Let [Formula: see text] be a path graph of [Formula: see text] vertices embedded in a metric space. We consider the problem of adding a new edge to [Formula: see text] so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of [Formula: see text]. Previously, the “continuous” version of the problem where a center may be a point in the interior of an edge of the graph was studied and a linear-time algorithm was known. Our “discrete” version of the problem has not been studied before. We present a linear-time algorithm for the problem.

We study a turn-based game in a simply connected polygonal environment [Formula: see text] between a pursuer [Formula: see text] and an adversarial evader [Formula: see text]. Both players can move in a straight line to any point within unit distance during their turn. The pursuer [Formula: see text] wins by capturing the evader, meaning that their distance satisfies [Formula: see text], while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader [Formula: see text] always knows the location of [Formula: see text]. Meanwhile, [Formula: see text] only has line-of-sight visibility: [Formula: see text] observes the evader’s position only when the line segment connecting them lies entirely within the polygon. Therefore [Formula: see text] must search for [Formula: see text] when the evader is hidden from view. We provide a winning strategy for [Formula: see text] in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line [Formula: see text] can be moved continuously over [Formula: see text] so that (1) [Formula: see text] is a line segment and (2) every point on the boundary [Formula: see text] is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook’s strategy during its pursuit phase, rather than the well-known lion’s strategy. The rook’s strategy is crucial for obtaining a capture time that is linear in the area of [Formula: see text]. For both monotone and scallop polygons, our algorithm has a capture time of [Formula: see text], where [Formula: see text] is the number of polygon vertices.

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.

In this paper, we study the problem of designing robust algorithms for computing the minimum spanning tree, the nearest neighbor graph, and the relative neighborhood graph of a set of points in the plane, under the Euclidean metric. We use the term “robust” to denote an algorithm that can properly handle degenerate configurations of the input (such as co-circularities and collinearities) and that is not affected by errors in the flow of control due to round-off approximations. Existing asymptotically optimal algorithms that compute such graphs are either suboptimal in terms of the arithmetic precision required for the implementation, or cannot handle degeneracies, or are based on complex data structures. We present a unified approach to the robust computation of the above graphs. The approach is a variant of the general region approach for the computation of proximity graphs based on Yao graphs, first introduced in Ref. 43 (A. C.-C. Yao, On constructing minimum spanning trees in [Formula: see text]-dimensional spaces and related problems, SIAM J. Comput. 11(4) (1982) 721–736). We show that a sparse supergraph of these geometric graphs can be computed in asymptotically optimal time and space, and requiring only double precision arithmetic, which is proved to be optimal. The arithmetic complexity of the approach is measured by using the notion of degree, introduced in Ref. 31 (G. Liotta, F. P. Preparata and R. Tamassia, Robust proximity queries: An illustration of degree-driven algorithm design, SIAM J. Comput. 28(3) (1998) 864–889) and Ref. 3 (J. D. Boissonnat and F. P. Preparata, Robust plane sweep for intersecting segments, SIAM J. Comput. 29(5) (2000) 1401–1421). As a side effect of our results, we solve a question left open by Katajainen27 (J. Katajainen, The region approach for computing relative neighborhood graphs in the [Formula: see text] metric, Computing 40 (1987) 147–161) about the existence of a subquadratic algorithm, based on the region approach, that computes the relative neighborhood graph of a set of points [Formula: see text] in the plane under the [Formula: see text] metric.

In 1982, Drezner proposed the [Formula: see text]-centroid problem on the plane, in which two players, called the leader and the follower, open facilities to provide service to customers in a competitive manner. The leader opens the first facility, and the follower opens the second. Customers will each patronize the facility closer to them (with ties broken in favor of the first one), thereby deciding the market share of the two facilities. The goal is to find the best position for the leader’s facility so that its market share is maximized. The best algorithm of this problem is an [Formula: see text]-time parametric search approach, which searches over the space of market share values. In the same paper, Drezner also proposed a generalized version of [Formula: see text]-centroid problem by introducing a minimal distance constraint [Formula: see text], such that the follower’s facility is not allowed to be located within a distance [Formula: see text] from the leader’s. He proposed an [Formula: see text]-time algorithm for this generalized version by identifying [Formula: see text] points as the candidates of the optimal solution and checking the market share for each of them. In this paper, we develop a new parametric search approach searching over the [Formula: see text] candidate points, and present an [Formula: see text]-time algorithm for this generalized version, thereby closing the [Formula: see text] gap between the two bounds.

Let [Formula: see text] be a set of stochastic sites, where each site is a tuple [Formula: see text] consisting of a point [Formula: see text] in [Formula: see text]-dimensional space and a probability [Formula: see text] of existence. Given a query point [Formula: see text], we define its most likely nearest neighbor (LNN) as the site with the largest probability of being [Formula: see text]’s nearest neighbor. The Most Likely Voronoi Diagram (LVD) of [Formula: see text] is a partition of the space into regions with the same LNN. We investigate the complexity of LVD in one dimension and show that it can have size [Formula: see text] in the worst-case. We then show that under non-adversarial conditions, the size of the [Formula: see text]-dimensional LVD is significantly smaller: (1) [Formula: see text] if the input has only [Formula: see text] distinct probability values, (2) [Formula: see text] on average, and (3) [Formula: see text] under smoothed analysis. We also describe a framework for LNN search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models as well as the worst-case with a bounded number of distinct probabilities. The Pareto-set framework is also applicable to multi-dimensional LNN search via reduction to a sequence of nearest neighbor and spherical range queries.

We study a modified version of the partial Fréchet similarity that is motivated by real world applications, e.g. the analysis of spectroscopic data in the context of astroinformatics and the analysis of birds’ migration trajectories. In those practical applications of curve matching it is often necessary to ignore outliers while dissimilarities regarding individual directions should be weighted by individual costs. We enable both by computing the partial Fréchet similarity between polygonal curves w.r.t. a non-uniform metric. In particular, we measure distances by a function [Formula: see text] that is induced by a set of weighted vectors. We discuss the approximation quality of [Formula: see text] regarding any [Formula: see text] metric and present a polynomial time algorithm for computing an exact solution of the resulting modified partial Fréchet similarity.

Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster’s magnitude of a metric space. Spread is generalized to infinite metric spaces equipped with a measure and is calculated for spheres and straight lines. For Riemannian manifolds the spread is related to the volume and total scalar curvature. A notion of scale-dependent dimension is introduced and seen for approximations to certain fractals to be numerically close to the Minkowski dimension of the original fractals.